Page 388 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 15] GAMMA AND BETA FUNCTIONS 379

ð 1
x 1 y 1
t dt
Bðx; yÞ¼ ð1 tÞ ð16Þ
0
If x 1 and y 1, this is a proper integral. If x > 0; y > 0 and either or both x < 1or y < 1, the
integral is improper but convergent.
It is shown in Problem 15.11 that the beta function can be expressed through gamma functions in the
following way
ðxÞ ð yÞ
Bðx; yÞ¼ ð17Þ
ðx þ yÞ
Many integrals can be expressed through beta and gamma functions. Two of special interest are
ð =2 1
sin 2x 1 cos 2y 1 d ¼ Bðx; yÞ¼ 1 ðxÞ ð yÞ ð18Þ
0 2 2 ðx þ yÞ
ð p 1
1 x
0 < p < 1 ð19Þ
dx ¼ ð pÞ ð p 1Þ¼
0 1 þ x sin p
See Problem 15.17. Also see Page 377 where a classical reference is given. Finally, see Chapter 16,
Problem 16.38 where an elegant complex variable resolution of the integral is presented.

DIRICHLET INTEGRALS
x p y q z r

If V denotes the closed region in the first octant bounded by the surface þ þ ¼ 1 and
a b c
the coordinate planes, then if all the constants are positive,

ðð ð
1 1
1 a b c p q r
x y z dx dy dz ¼ ð20Þ
pqr
V 1 þ þ þ
p q r
Integrals of this type are called Dirichlet integrals and are often useful in evaluating multiple
integrals (see Problem 15.21).

Solved Problems

THE GAMMA FUNCTION

15.1. Prove: (a) ðx þ 1Þ¼ x ðxÞ; x > 0; ðbÞ ðn þ 1Þ¼ n!; n ¼ 1; 2; 3; .. . .
M
ð ð
1 v x v x
x e dx ¼ lim x e dx
ðaÞ ðv þ 1Þ¼
0 M!1 0
ð M
x
v
x M
¼ lim ðx Þð e Þj 0 ð e Þðvx v 1 Þ dx
M!1 0
M
ð
v M
¼ lim M e þ v x v 1 x dx ¼ v ðvÞ if v > 0
e
M!1 0
ð ð M
1
e x dx ¼ lim e x dx ¼ lim ð1 e M Þ¼ 1:
ðbÞ ð1Þ¼
0 M!1 0 M!1
Put n ¼ 1; 2; 3; .. . in ðn þ 1Þ¼ n ðnÞ. Then
ð2Þ¼ 1 ð1Þ¼ 1; ð3Þ¼ 2 ð2Þ¼ 2 1 ¼ 2!; ð4Þ¼ 3 ð3Þ¼ 3 2! ¼ 3!
In general, ðn þ 1Þ¼ n! if n is a positive integer.